Using closed orbits to bifurcate many periodic solutions for pendulum-type equations

For pendulum-like equations, perturbation-type arguments and topological tools provide the existence of external forces with many associated periodic solutions.     

On periodic solutions of even-order ordinary differential equations

Optimal in a certain sense sufficient conditions are given for the existence and uniqueness of ω-periodic solutions of the nonautonomous ordinary differential equation u ^(2m) =f(t,u,...,u ^(m-1) ), where the function f:ℝ×ℝ ^m →ℝ is periodic with respect to the first argument with period ω. Received: December 21, 1999; in final form: August 12, 2000?Published online: October 2, 2001     

Numerical computation of branch points in ordinary differential equations

Summary This paper deals with the computation of branch points in ordinary differential equations. A direct numerical method is presented which requires the solution of only one boundary value problem. The method handles the general case of branching from a nontrivial solution which is a-prioriunknown. A testfunction is proposed which may indicate branching if used in continuation methods. Several real-life problems demonstrate the procedure.     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China     

First order ordinary differential equations with several periodic solutions

The method of upper and lower solutions and convexity arguments are used to prove sharp results for the existence and multiplicity of periodic solutions for first order ordinary differential equations depending upon a parameter.Dedicated to Professor H. W. Knobloch for his sixtieth birthday     

Multiple periodic solutions of ordinary differential equations with double resonance

In this paper we study by Morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin.     

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

Existence of nontrivial periodic solutions of certain nonautonomous ordinary differential equations

Degree theory is used to establish sufficient conditions for the existence of a non-trivial periodic solution of the vector differential equation f(D)x + BMg(D)x = 0, which generalises the feedback control equation. These conditions require that the matrix M should satisfy two elliptic ball criteria, one of which is relevant near the origin of the phase space and the other near the point at infinity.     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

有序Banach空间常微分方程的正周期解

依据凝聚锥映射的一个krasnoselskii型不动点定理,在有序Banach空间中获得了二阶常微分方程．-u^n(t) Mu(t)＝f(t,u(t))正ω—周期解的存在性结果。     

Positive periodic solutions of singular systems of first order ordinary differential equations

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

Rigorous computational shadowing of orbits of ordinary differential equations

Summary. The existence of a true orbit near a numerically computed approximate orbit -- shadowing -- of autonomous system of ordinary differential equations is investigated. A general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the approximate orbit in terms of computable quantities, is proved. The practical use and the effectiveness of this theorem is demonstrated in the numerical computations of chaotic orbits of the Lorenz equations. Received December 15, 1993